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Rational points near manifolds and Khintchine theorem

Victor Beresnevich, Shreyasi Datta

TL;DR

This work resolves Khintchine-type questions for arbitrary nondegenerate manifolds in $\mathbb{R}^n$, removing analyticity constraints and extending results to inhomogeneous settings and Hausdorff measures (Jarník-type refinements). Central to the advance is a sharpened quantitative nondivergence framework (BKM_new) and a duality argument, enabling both lower and upper bounds on rational points near manifolds and a full divergence/convergence theory for Hausdorff measures. The paper also analyzes the spectrum of inhomogeneous Diophantine exponents, with explicit results on Veronese curves, and provides a detailed decomposition into generic and nongeneric parts to prove convergence. Collectively, these results give a comprehensive, high-dimensional understanding of Diophantine approximation on manifolds, with broad implications for metric number theory and fractal geometry on manifolds.

Abstract

In this paper, we complete the long-standing challenge to establish a Khintchine-type theorem for arbitrary nondegenerate manifolds in $\mathbb{R}^n$. In particular, our main result finally removes the analyticity assumption from the Khintchine type theorem proved in [Ann. of Math. 175 (2012), 187-235]. Furthermore, we obtain a Jarník-type refinement of our main result, which uses Hausdorff measures. The results are also obtained in the inhomogeneous setting. The proofs are underpinned by a sharper version of a `quantitative nondivergence' result of Bernik--Kleinbock--Margulis and a duality argument which we use to study rational points near manifolds.

Rational points near manifolds and Khintchine theorem

TL;DR

This work resolves Khintchine-type questions for arbitrary nondegenerate manifolds in , removing analyticity constraints and extending results to inhomogeneous settings and Hausdorff measures (Jarník-type refinements). Central to the advance is a sharpened quantitative nondivergence framework (BKM_new) and a duality argument, enabling both lower and upper bounds on rational points near manifolds and a full divergence/convergence theory for Hausdorff measures. The paper also analyzes the spectrum of inhomogeneous Diophantine exponents, with explicit results on Veronese curves, and provides a detailed decomposition into generic and nongeneric parts to prove convergence. Collectively, these results give a comprehensive, high-dimensional understanding of Diophantine approximation on manifolds, with broad implications for metric number theory and fractal geometry on manifolds.

Abstract

In this paper, we complete the long-standing challenge to establish a Khintchine-type theorem for arbitrary nondegenerate manifolds in . In particular, our main result finally removes the analyticity assumption from the Khintchine type theorem proved in [Ann. of Math. 175 (2012), 187-235]. Furthermore, we obtain a Jarník-type refinement of our main result, which uses Hausdorff measures. The results are also obtained in the inhomogeneous setting. The proofs are underpinned by a sharper version of a `quantitative nondivergence' result of Bernik--Kleinbock--Margulis and a duality argument which we use to study rational points near manifolds.
Paper Structure (23 sections, 28 theorems, 151 equations)

This paper contains 23 sections, 28 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Nondegeneracy of maps and manifolds
  3. Rational points near manifolds
  4. Hausdorff dimension and measure results
  5. Spectrum of inhomogeneous exponents
  6. Sharp Quantitative Nondivergence
  7. Proof of the Claim
  8. Auxiliary statements and definitions
  9. Some geometry of numbers
  10. Further assumptions and notation
  11. Lower bounds
  12. Proof of Theorem \ref{['mainthm: lower bound']}
  13. Proof of Theorem \ref{['main: Haus_div']}
  14. Proof of divergence part in Theorem \ref{['thm: main']}
  15. Upper bounds
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $n\ge2$, $\psi$ be non-increasing, $\boldsymbol{\theta}\in\mathbb{R}^n$ and $\mathcal{M}$ be any nondegenerate submanifold of $\mathbb{R}^n$. Then, almost no/almost every point $\mathbf{x}\in\mathcal{M}$ is $(\psi,\boldsymbol{\theta})$-approximable provided that the sum converges/diverges.

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2
  • Theorem 1.3: Lower bound
  • Theorem 1.4: Upper bound
  • Theorem 2.1: Divergence case
  • Remark 2.1
  • Theorem 2.2: Convergence case
  • ...and 30 more